Finite Group DatabasebetaFinite Group Databasebeta| order | 1312 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C41 · Schur multiplier (H2) = C22 |
| direct product | #135×C41 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C41 | D4 |
| commutator, Frattini, cosocle | C4 | C23×C41 |
| socle | C2×C41 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C41 ◁ C2×C4×C41 ◁ G | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 656 |
A minimal faithful permutation representation — 3 generators on 57 points:
(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)574 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1312, 42);